Answer Happy

1. Let N ∼ Poi(4) and Xi ∼ Bin(2, 1 2 ), for i = 1, 2, . . . The
random variables N, X1, X2, . . . are independent. Define Y = X N
i=1 Xi , where Y = 0 if N = 0.
(a) Show that GY (z) = E(z Y ) = exp
(b) Determine P(Y = 0) and P(Y = 1). [2 marks]
(c) Determine EY and Var(Y ). [2 marks]
(d) Let V1 and V2 be two independent random variables with Vi ∼
Poi(λi), i = 1, 2. Show that there exists λ1, λ2 > 0 such that
V1 + 2V2 has the same distribution as Y . [3 marks]
= 1. Let N ~ Poi(4) and X; ~ Bin(2, ), for i = 1, 2, ... The random variables N, X1, X2, ... are independent. Define N Y - ΣΧ;, where Y = 0 if N = 0. = i=1 = = = (a) Show that Gy(z) = E(zY) = exp (22 + 2z – 3), ZER. [3 marks] (b) Determine P(Y = 0) and P(Y = 1). [2 marks] (c) Determine EY and Var(Y). [2 marks] (d) Let Vi and V2 be two independent random variables with Vi ~ Poi(li), i = 1,2. Show that there exists 11, 12 > 0 such that V1 + 2V2 has the same distribution as Y. [3 marks] =